Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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July 24, 2023 at 9:14am

#1052996

Millimeters of Mercury

There are several articles in my queue, from various sources, about planets in our solar system. Fittingly, though randomly, the one about Mercury (from Vox) shows up here first.

Mercury was a total mystery for centuries. We're finally learning its secrets.

It's still a mystery to me. I've never knowingly seen it in the sky.

All the other visible planets, sure, no problem. Venus, when it's visible at all, is very visible, usually the brightest thing in the night sky apart from the Moon. Mars is generally easy to spot for its color; Jupiter, like Venus, tends to be bright, but isn't limited to just after sunset or just before sunrise. Saturn, while you can't see its rings with unaided eyes, is generally recognizable.

I've even seen the next planet out, through a telescope that someone else pointed. No, I won't name it until its article comes up, lest someone make a worn-out, juvenile joke about it.

But Mercury? Its visibility is even more limited than that of Venus, always close to the twilight horizon if it's visible at all, not to mention much dimmer, and that makes it tough if you don't know exactly what to look for. And despite my interest in astronomy, I never could be arsed to get one of those cellphone apps that tells you what everything is.

It's entirely possible that I've seen it and mistaken it for a star near the horizon, where some constellational context is missing.

Anyway, that's not what the article is about. Thanks to space robots (we have SPACE ROBOTS), we've seen Mercury quite a bit.

On Friday, the US Geological Survey released the most comprehensive topographical map of Mercury ever created, depicting its craters, ridges, volcanoes, and mountains — some rising more than 2 miles high — in fine detail:

You'll have to go to the link to see the map. Also, "on Friday" refers to a long-ago Friday in early 2016, as the article is pretty old now.

Because Mercury is so small and close to the sun, it's tricky to send spacecraft to visit it — before NASA's MESSENGER probe, the only craft that had come close was Mariner 10, which made a series of quick flybys in the 1970s.

The difficulty of getting to Mercury may seem counterintuitive. Woudn't a probe just fall toward the Sun? Well, no, for the same reason Earth (much to our relief) doesn't; you have to cancel a lot of angular momentum, and that takes fuel, careful planning, and lots and lots of math.

So, the rest of the article consists of some of the stuff we managed to figure out thanks to fuel, careful planning, and lots and lots of math.

1) Mercury is shrinking

And no, not because of #2 here:

2) Mercury has water ice

This has long been theorized, but lots of things were theorized that turned out not to be the case. The brief explanation for how ice can exist on the surface of a planet that's famously close to a giant perpetual fusion bomb is that Mercury has no real atmosphere to distribute heat, and the ice is in shadowed craters at the poles.

3) Mercury had a violent, volcanic past

While, again, it's good to get evidence, this isn't exactly surprising given that other terrestrial planets (and the Moon) have or had volcanic activity.

4) We can't quite figure out how Mercury formed

To be fair, we're a little fuzzy on the details of how any planets formed.

What's cool is that we're able to figure out things like how big a planet's core is and what it's made of, and that we have the technology to send robots there in the first place.

5) Mercury has a weird, off-center magnetic field

"Weird" meaning "No, we don't understand that yet, either, but it'll help us do more science."

So, there it is: stuff we didn't know before, or suspected but didn't have proof of. Always more to find out.

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